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Scattering theory for the defocusing fourth-order Schrödinger equation

In this paper, we study the global well-posedness and scattering theory for the defocusing fourth-order nonlinear Schrödinger equation (FNLS) $iu_t+Δ^2 u+|u|^pu=0$ in dimension $d\geq9$. We prove that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $u\in L_t^\infty(I;\dot H^{s_c}_x(\R^d))$ with all $s_c:=\frac{d}2-\frac4p\geq1$ if $p$ is an even integer or $s_c\in[1,2+p)$ otherwise, then $u$ is global and scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical and energy-subcritical nonlinear Schrödinger equation (NLS) and nonlinear wave equation (NLW). We will give a uniform way to treat the energy-subcritical, energy-critical and energy-supercritical FNLS, where we utilize the strategy derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to exclude the existence of three scenarios: finite time blowup; soliton-like solution and low to high frequency cascade. Making use of the No-waste Duhamel formula, we deduce that the energy or mass of the finite time blow-up solution is zero and so get a contradiction. Finally, we adopt the double Duhamel trick, the interaction Morawetz estimate and interpolation to kill the last two scenarios.